Problem 40 - Calculus: Early Transcendentals

Question

Horizontal asymptotes Determine \(\lim _{x \rightarrow \infty} f(x)\) and \(\lim _{x \rightarrow-\infty} f(x)\) for the following functions. Then give the horizontal asymptotes of \(f(\text {if any})\). $$f(x)=\frac{12 x^{8}-3}{3 x^{8}-2 x^{7}}$$

Step-by-Step Solution

Answer

Answer: The horizontal asymptote of the function is at \(y=4\).

Step 1: Find the coefficients of the highest power term

The given function has the highest power term x^8 in both numerator and denominator. Identify the coefficients in front of these terms: $$ f(x) = \frac{12x^8 - 3}{3x^8 - 2x^7} = \frac{12}{3} \cdot \frac{x^8 - \frac{1}{4} \cdot x^0}{x^8 - \frac{2}{3} \cdot x^0} $$

Step 2: Evaluate the limits by dividing all terms

Now, consider the limits as x approaches infinity and negative infinity: $$ \lim_{x\to\infty} f(x) = \lim_{x\to\infty} \frac{12}{3} \cdot \frac{x^8 - \frac{1}{4} \cdot x^0}{x^8 - \frac{2}{3} \cdot x^0} = \frac{12}{3} $$ $$ \lim_{x\to-\infty} f(x) = \lim_{x\to-\infty} \frac{12}{3} \cdot \frac{x^8 - \frac{1}{4} \cdot x^0}{x^8 - \frac{2}{3} \cdot x^0} = \frac{12}{3} $$

Step 3: Determine the horizontal asymptotes

From the evaluated limits, we can conclude the horizontal asymptotes of the function are: $$ y = \lim_{x\to\infty} f(x) = 4 $$ and $$ y = \lim_{x\to-\infty} f(x) = 4 $$ Thus, the horizontal asymptote is at \(y=4\).

Key Concepts

Limits at Infinity

Understanding limits at infinity is crucial for analyzing the behavior of functions as they approach very large positive or negative values. When we calculate the limit of a function as \(x\) approaches infinity (\(x \to \infty\)) or negative infinity (\(x \to -\infty\)), we are essentially trying to determine what value the function \(f(x)\) gets closer to, even though \(x\) itself is increasing or decreasing without bound.

In the original exercise, we find \(\lim_{x \rightarrow \infty} f(x)\) and \(\lim_{x \rightarrow -\infty} f(x)\) for the function \(f(x) = \frac{12x^8 - 3}{3x^8 - 2x^7}\). By focusing on the highest power terms in both the numerator and the denominator, we simplify the function to analyze the limit efficiently. The terms \(12x^8\) and \(3x^8\) dominate the behavior as \(x\) approaches infinity. Therefore, the limits simplify to analyzing the fraction \(\frac{12}{3}\), resulting in both limits equalling 4.

This understanding of limits helps identify horizontal asymptotes, which indicate a boundary that \(f(x)\) will approach but never fully reach in the function's graph.

Rational Functions

Rational functions, composed of a polynomial in the numerator and a polynomial in the denominator, hold interesting traits, especially when it comes to limits. In our provided function, \(f(x) = \frac{12x^8 - 3}{3x^8 - 2x^7}\), we look at how these polynomials interact.

The degree of the polynomial (the highest power of \(x\)) in both the numerator and the denominator significantly affects the function's behavior as \(x\) approaches infinity. Generally, if the degree of the numerator is equal to the degree of the denominator, as is the case here, the horizontal asymptote can be found by looking at the ratio of the leading coefficients (here, \(\frac{12}{3} = 4\)).

This type of simplification is incredibly useful because it lets us quickly identify the function's behavior at extremes without evaluating every term. Rational functions are particularly amicable to this kind of analysis owing to their polynomial nature allowing straightforward simplification by looking at the highest degree terms.

Asymptotic Behavior

Asymptotic behavior is the study of how functions behave as inputs become very large or very small. This concept is closely linked with limits at infinity and horizontal asymptotes. In the context of rational functions, understanding this behavior often revolves around identifying horizontal, vertical, or oblique asymptotes—if any.

In analyzing our function \(f(x) = \frac{12x^8 - 3}{3x^8 - 2x^7}\), we found that the function approaches a horizontal line \(y = 4\) as \(|x|\) (i.e., \(x\) as it tends to \(\infty\) or \(-\infty\)) grows. This means that no matter how large \(x\) gets, \(f(x)\) will get arbitrarily close to the line \(y = 4\), but not cross it.

Horizontal asymptotes like this can help sketch the overall trend of the function graph and provide insights into how the function behaves at its extremes. They symbolize a leveling-off of the function values, insinuating stability or long-term behavior of the graph.